Matrices and determinants are used to perform various arithmetic operations involving an array of elements. Matrices are a rectangular array of elements that are represented in the form of rows and columns. And determinants are calculated for a matrix and it is a single numeric value that has been computed from this array of elements. The matrix is represented with an alphabet in upper case and is written as A, and the determinant is represented as |A|. Matrices and determinants have differences in their properties. The multiplication of a constant K with a matrix multiplies every element of the matrix, and the multiplication of a constant K with a Determinant multiplies with the elements of any particular row or columns. Let us learn more about the properties, and differences between matrices and determinants with the help of examples, FAQs.

---

A system of linear equations is a group of two or more linear equations having the same variables. For example, x + 2y = 14 , 2x + y = 6. To compare equations in linear systems, the best way is to see how many solutions both equations have in common. If there is nothing common between the two equations then it can be called inconsistent. But it will be called consistent if anyone ordered pair can solve both the equations. If the equation carries more than one point in common then it will be called dependent. But what does ‘solution in common’ mean? It means that if there is at least one ordered pair that can solve both the equations in spite of having many equations that do not.

---

A matrix is the rectangular arrangement of numbers, symbols, or characters representing a set of data in any system. The elements of a matrix are arranged in rows and columns. Order of a matrix is the representation of its rows and columns number in the form MxN where m is the number of rows and n is the number of columns. Two matrices are said to be equal if they have the same order and their elements are identical. There is a difference between the terms ‘equal matrices’ and ‘equivalent matrices. Equivalence of two matrices is denoted using the symbol ‘~’. Two matrices are said to be equivalent if one matrix can be modified through the elementary transformation of the matrix to obtain the other matrix.

---

In this section, we learn to solve systems of linear equations using a process called the Gauss-Jordan method. The process begins by first expressing the system as a matrix, and then reducing it to an equivalent system by simple row operations. The process is continued until the solution is obvious from the matrix. The matrix that represents the system is called the augmented matrix, and the arithmetic manipulation that is used to move from a system to a reduced equivalent system is called a row operation.

---

Complex numbers are helpful in finding the square root of negative numbers. The concept of complex numbers was first referred to in the 1st century by a greek mathematician, Hero of Alexandria when he tried to find the square root of a negative number. But he merely changed the negative into positive and simply took the numeric root value. Further, the real identity of a complex number was defined in the 16th century by Italian mathematician Gerolamo Cardano, in the process of finding the negative roots of cubic and quadratic polynomial expressions. Complex numbers have applications in much scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and vibration analysis. Here we can understand the definition, terminology, visualization of complex numbers, properties, and operations of complex numbers.

---

Every complex number has another complex number associated with it, known as the complex conjugate. A complex conjugate of a complex number is another complex number that has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign. The product of a complex number and its complex conjugate is a real number. A complex conjugate gives the mirror image of the complex number about the horizontal axis (real axis) in the Argand plane. In this article, we will explore the meaning of conjugate of a complex number, its properties, complex root theorem, and some applications of the complex conjugate.

---

The modulus of a complex number is the square root of the sum of the squares of the real part and the imaginary part of the complex number. If z is a complex number, then the modulus of the complex number z is given by, √{[Re(z)]2 + [Im(z)]2} and it is denoted by |z|. The modulus of complex number z = a + ib is the distance between the origin (0, 0) and the point (a, b) in the complex plane. Since the modulus of a complex number is the distance, its value is always non-negative.

---

---

The square root of complex number gives a pair of complex numbers whose square is the original complex number. The square root of a complex number can be determined using a formula. Just like the square root of a natural number comes in pairs (Square root of x2 is x and -x), the square root of complex number a + ib is given by √(a + ib) = ±(x + iy), where x and y are real numbers. In this article, let us go through the concept of finding the square root of a complex number using the assumption method and using the formula. We will also learn to determine the square root of a complex number in polar form along with some solved examples for a better understanding of the concept.

---

roots are the imaginary roots of quadratic equations which have been represented as complex numbers. The square root of a negative number is not possible and hence we transform it into a complex number. The quadratic equations having discriminant values lesser than zero b2 - 4ac < 0, is transformed using i2 = -1, to obtain the complex roots. Here -D is written as i2D.

---

roots are the imaginary roots of quadratic equations which have been represented as complex numbers. The square root of a negative number is not possible and hence we transform it into a complex number. The quadratic equations having discriminant values lesser than zero b2 - 4ac < 0, is transformed using i2 = -1, to obtain the complex roots. Here -D is written as i2D.

---

Polynomials are one of the significant concepts of mathematics, and so are Polynomial Equations, where the relation between numbers and variables is explained in a pattern. In math, there are a variety of equations formed with algebraic expressions. Polynomial Equations are also a form of algebraic equations. Let us learn more about polynomial equations along with their types and the process of solving them.

---

The Theory of equations is essentially the study of the inter-relationship between an equation and the polynomials that comprise the equation. An equation is basically a mathematical expression, equating two different quantities, values, or expressions. The main issue in this field of study is that an algebraic equation will have an algebraic solution. To put it simply, the idea of complex solutions was not common knowledge. Therefore, finding the solution to a single unknown variable in a non-linear polynomial equation was a stumbling block. Mathematician Evariste Galois was the first to solve this problem in the year 1830. His theory gave a distinct outline to help understand which equations could be solved by radical values. This is how the theory of equations became its own area rather than being synonymous with algebra itself.

---

Inverse trigonometric functions, as a topic of learning, are closely related to the basic trigonometric functions. The domain and the range of the trigonometric functions are converted to the range and domain of the inverse trigonometric functions. In trigonometry, we learn about the relationships between angles and sides in a right-angled triangle. Similarly, we have inverse trigonometry functions. The basic trigonometric functions are sin, cos, tan, cosec, sec, and cot. The inverse trigonometric functions on the other hand are denoted as sin-1x, cos-1x, cot-1 x, tan-1 x, cosec-1 x, and sec-1 x. Inverse trigonometric functions have all the formulas of the basic trigonometric functions, which include the sum of functions, double and triple of a function. Here we shall try to understand the transformation of the trigonometric formulas to inverse trigonometric formulas.

---

Principal values of trigonometric functions are solutions of trigonometric functions, for which the θ value lies between 0 < θ < 2π. The value of the trigonometric function repeat after an interval of 2π, and the values lesser than 2π are the principal values of trigonometric functions. Let us learn more about the method to find the principal value of the trigonometric function, with examples, solutions.

---

Conic sections or sections of a cone are the curves obtained by the intersection of a plane and cone. There are three major sections of a cone or conic sections: parabola, hyperbola, and ellipse(the circle is a special kind of ellipse). A cone with two identical nappes is used to produce the conic sections. All the sections of a cone or conic sections have different shapes, but they do share some common properties which we will read in the following sections. Let us check the conic section formulas, conic equations and its parameters, with examples, FAQs.

---

Conic sections are the curves obtained when a plane cuts the cone. A cone generally has two identical conical shapes known as nappes. We can get various shapes depending upon the angle of the cut between the plane and the cone and its nappe. By cutting a cone by a plane at different angles,

---

The dot product is one way of multiplying two or more vectors. The resultant of the dot product of vectors is a scalar quantity. Thus, the dot product is also known as a scalar product. Algebraically, it is the sum of the products of the corresponding entries of two sequences of numbers. Geometrically, the dot product of two vectors is the product of their Euclidean magnitudes and the cosine of the angle between them. The dot product of vectors finds various applications in geometry, mechanics, engineering, and astronomy. Let us discuss the dot product in detail in the upcoming sections.

---

Distance between two lines means how far the two lines are located from each other. A line is a figure that is formed when two points are connected with minimum distance between them, and both the ends of a line are extended to infinity. The distance between two lines can be calculated by measuring the perpendicular distance between them. Generally, we find the distance between two parallel lines. Also, for two non-intersecting lines which are lying in the same plane, the shortest distance between them is the distance that is the shortest of all the distances between two points lying on both lines. Let us learn more about the distance between two lines along with a few solved examples and practice questions.

---

Vectors algebra involves algebraic operations across vectors. The algebraic operations involving the magnitude and direction of vectors is performed in vector algebra. Vector algebra helps for numerous applications in physics, and engineering to perform addition and multiplication operations across physical quantities, represented as vectors in three-dimensional space. Let us learn more about vector algebra, operations in vector algebra, vector types, with the help of solved examples, and practice questions.

---

Torque is defined as the measure of the force that can cause an object to rotate about an axis. Force is the thing that causes an object to accelerate in linear kinematics. Like in that way, torque causes an angular acceleration. So, torque can be defined as the rotational equivalent of linear force. And the point where the object rotates is called the axis of rotation. Basically, according to physics, torque is simply the tendency of a force to turn or twist. Different terms like moment or moment of force are widely used to describe torque. Generally, torque refers to the twisting force that causes motion and it also refers to the turning effect. Basically, the point of the rotation of the object is called the axis of rotation. Everyone is making use of this force without realizing this fact. And also the Torque is a vector quantity. So, the direction of the torque vector depends on the direction of the force on the axis. Torque can either be defined as static or dynamic.

---

Differential calculus studies the rate of change of two quantities. Calculus can be divided into two parts, namely, differential calculus and integral calculus. In differential calculus, the derivative equation is used to describe the rate of change of a function whereas in integral calculus the area under a curve is studied. One of the main uses of differential calculus is in finding the minimum or maximum value of a given function as part of an optimization problem. In this article, we will learn more about differential calculus, the important formulas, and various associated examples.

---

Tangents and Normals are the lines associated with curves such as a circle, parabola, ellipse, hyperbola. A tangent is a line touching the curve at one distinct point, and this distinct point is called the point of contact. Normal is a line perpendicular to the tangent, at the point of contact. The normal is also passing through the focus of the curve. There are numerous tangents that can be drawn to a curve, at each of the distinct points lying on the curve. The tangents and Normals are straight lines and hence they are represented as a linear equation in x and y. The general form of the equation of a tangent and normal is ax + by + c = 0. The point of contact satisfies the equation of the tangent and the equation of the curve.

---

The cartesian form helps to represent geometric entities in the cartesian plane. A point, a line, or a plane can be easily represented in a three-dimensional plane, across the x-axis, y-axis, z-axis, in cartesian form. The cartesian form of representation of a point is (x, y, z), the line is (x - x1)/a = (y - y1)/b = (z - z1)/c, and the plane is ax + by + cz = d. The cartesian form is helpful to represent the geometric entities as algebraic expressions in three-dimensional geometry. Let us learn more about the conversion of cartesian form to vector form, the difference between cartesian form and vector form, with the help of examples, FAQs.

---

An equation that contains the derivative of an unknown function is called a differential equation. The rate of change of a function at a point is defined by the derivatives of the function. A differential equation relates these derivatives with the other functions. Differential equations are mainly used in the fields of biology, physics, engineering, and many. The main purpose of the differential equation is for studying the solutions that satisfy the equations and the properties of the solutions. Let us discuss the definition, types, methods to solve the differential equation, order, and degree of the differential equation, types of differential equations, with real-world examples, and practice problems.

---

---

There is a number of methods of calculations among which are functions, differentiation, and integration. Applications of Integrals are applied in various fields like Mathematics, Science, Engineering. Further, for the calculation of areas or irregular shapes in a two-dimensional space, we use majorly integrals formulas. Here a brief introduction on integrals is given, with applications of integrals to find areas under simple curves, areas bounded by a curve and a line and area between two curves, and also the application of integrals in other mathematical disciplines along with the solved examples.

---

An ordinary differential equation contains the derivative of an unknown function. The ordinary differential equation is an equation having variables and a derivative of the dependent variable with reference to the independent variable. The two types of ordinary differential equations are the homogeneous differential equation and non-homogeneous differential equation. Let us learn more about the ordinary differential equations along with the process of finding their order, degree, and solution.

---

The "tangent line" is one of the most important applications of differentiation. The word "tangent" comes from the Latin word "tangere" which means "to touch". The tangent line touches the curve at a point on the curve. So to find the tangent line equation, we need to know the equation of the curve (which is given by a function) and the point at which the tangent is drawn. Let us see how to find the slope and equation of the tangent line along with a few solved examples. Also, let us see the steps to find the equation of the tangent line of a parametric curve and a polar curve.

---

A continuous distribution has an infinite range of values. This means that the number of possible outcomes is uncountable, as we often see in variables such as time or temperature. Time is uncountable because it is infinite: you could count from 0 seconds to a trillion seconds…and keep on counting forevermore. Even if you limit time to a range, say between 0 and 2 seconds, it’s still uncountable because of the infinite number of possible divisions, such as 0.2 seconds, 0.0022 second, 0.00004 seconds, and so on. A continuous distribution also follows a smooth curve, which is why it is sometimes referred to as a smooth distribution.

---

Probability distribution is a function that gives the relative likelihood of occurrence of all possible outcomes of an experiment. There are two important functions that are used to describe a probability distribution. These are the probability density function or probability mass function and the cumulative distribution function. In statistics, there can be two types of data, namely, discrete and continuous. Based on this, a probability distribution can be classified into a discrete probability distribution and a continuous probability distribution. In this article, we will learn more about probability distribution and the various aspects that are associated with it.

---

Discrete Mathematics is a branch of mathematics that is concerned with “discrete” mathematical structures instead of “continuous”. Discrete mathematical structures include objects with distinct values like graphs, integers, logic-based statements, etc. In this tutorial, we have covered all the topics of Discrete Mathematics for computer science like set theory, recurrence relations, group theory, and graph theory.

---

A connector or logical connective is an element of language that allows us to construct new statements from existing ones. We can also consider them as logical operators, since we perform certain operations between propositions to construct new ones. The propositions obtained are compounds and can also be combined through connectives to form others.

---